Video Transcript
If the determinant of the
two-by-two matrix 𝑥, four, four, 𝑦 is equal to zero; the determinant of the
two-by-two matrix 𝑦, nine, nine, 𝑧 is equal to zero; and the determinant of the
two-by-two matrix 𝑥, one, one, 𝑧 is equal to zero, find the determinant of the
three-by-three matrix 𝑥, one, two, zero, 𝑦, three, zero, zero, 𝑧.
In this question, we’re told that
the determinant of three two-by-two matrices is equal to zero. And each of these three matrices
involves three unknowns: 𝑥, 𝑦, and 𝑧. We need to use this information to
determine the determinants of a three-by-three matrix involving the three
unknowns. So to answer this question, we’re
going to need to start by finding an expression for the determinants of the
three-by-three matrix. And we might be tempted to do this
by expanding over the first row. However, there’s a much simpler
method.
We need to notice that this matrix
is an upper triangular matrix. All of the entries below the main
diagonal are zero. We can then evaluate this
determinant by recalling one of the properties of the determinant, which says if
we’re trying to find the determinant of a square triangular matrix, then this is
just equal to the product of all of the entries on the main diagonal. Therefore, since this is an upper
triangular matrix, the determinant of this matrix is the product of the entries on
its main diagonal: 𝑥 times 𝑦 times 𝑧.
Therefore, to answer this question,
we need to determine the value of 𝑥, 𝑦, and 𝑧 from the three given
determinants. And to do this, we’re going to need
to evaluate each of the three determinants. Let’s start with the first
determinant. We want to evaluate the determinant
of a two-by-two matrix. And we do this by finding the
difference in the products of its diagonals. The determinant of this matrix is
𝑥 times 𝑦 minus four times four. And we can then simplify this since
four times four is 16. The determinant of this matrix is
𝑥𝑦 minus 16. But remember, we’re told in the
question the determinant of this matrix is equal to zero. Therefore, we have that zero is
equal to 𝑥𝑦 minus 16. We can find an expression for 𝑥𝑦
by adding 16 to both sides of the equation. We have that 𝑥𝑦 is equal to
16.
Let’s now apply this same process
to the second determinant. First, we evaluate the determinant
of this matrix by taking the difference in the products of the diagonals. That’s 𝑦 times 𝑧 minus nine times
nine. And then since nine times nine is
81, this simplifies to give us 𝑦𝑧 minus 81. And remember, we’re told in the
question this determinant is equal to zero. So we can set this equal to zero
and then add 81 to both sides of the equation. We get that 𝑦 times 𝑧 is equal to
81.
We now need to apply this process
one final time to the third and final determinant. First, we take the difference in
the products of the diagonals. The determinant of this matrix is
𝑥 times 𝑧 minus one times one, which simplifies to give us 𝑥𝑧 minus one. And we know this determinant is
equal to zero. We can then add one to both sides
of the equation to determine that 𝑥 times 𝑧 must be equal to one.
And at this point, we can notice
something interesting. We have three equations involving
our variables 𝑥, 𝑦, and 𝑧. And the left-hand side of each of
these three equations appears in our expression for the determinant. So we can try and find an
expression for this determinant by taking the product of each of these three
equations. First, taking the product of the
left-hand side of these three equations, we get 𝑥𝑦 times 𝑦𝑧 times 𝑥𝑧. This will then be equal to the
product of the right-hand side of each equation: 16 times 81 times one.
Let’s now simplify this
equation. First, on the left-hand side of the
equation, we have two factors of 𝑥, two factors of 𝑦, and two factors of 𝑧. So this simplifies to give us 𝑥
squared times 𝑦 squared times 𝑧 squared. Next, we can simplify the
right-hand side of the equation. 16 times 81 times one is 1,296. We’re now almost ready to find the
value of this determinant. We just need to use our laws of
exponents to take the constant exponent of two outside of the expression. 𝑥 squared times 𝑦 squared times
𝑧 squared is 𝑥 times 𝑦 times 𝑧 all squared. And this is the square of the
expression we want to find the value of. So we have 𝑥 times 𝑦 times 𝑧
squared is equal to 1,296.
We can find the value of 𝑥 times
𝑦 times 𝑧 by taking the square root of both sides of the equation. We get a positive and a negative
root. We get 𝑥 times 𝑦 times 𝑧 is
equal to positive or negative the square root of 1,296. And we can then evaluate this. We get positive or negative 36,
which is our final answer. The determinant of the given
three-by-three matrix is either equal to 36 or negative 36.
